On Topological String Theory with Calabi-Yau Backgrounds

Transcription

1 On Topological String Theory with Calabi-Yau Backgrounds Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität zu Bonn vorgelegt von Babak Haghighat aus Bonn Bonn 2009

2 2 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn. Referent: Prof. Dr. Albrecht Klemm Korreferent: Prof. Dr. Hans-Peter Nilles Tag der Promotion:

3 3 Abstract String theory represents a unifying framework for quantum field theory as well as for general relativity combining them into a theory of quantum gravity. The topological string is a subsector of the full string theory capturing physical amplitudes which only depend on the topology of the compactification manifold. Starting with a review of the physical applications of topological string theory we go on to give a detailed description of its theoretical framework and mathematical principles. Having this way provided the grounding for concrete calculations we proceed to solve the theory on three major types of Calabi-Yau manifolds, namely Grassmannian Calabi-Yau manifolds, local Calabi-Yau manifolds, and K3 fibrations. Our method of solution is the integration of the holomorphic anomaly equations and fixing the holomorphic ambiguity by physical boundary conditions. We determine the correct parameterization of the ambiguity and new boundary conditions at various singularity loci in moduli space. Among the main results of this thesis are the tables of degeneracies of BPS states in the appendices and the verification of the correct microscopic entropy interpretation for five dimensional extremal black holes arising from compactifications on Grassmannian Calabi-Yau manifolds. 3

4 4 Ich möchte mich zuallererst bei Prof. Albrecht Klemm dafür bedanken, dass er es mir ermöglicht hat, meine Doktorarbeit auf diesem für mich besonders spannenden Gebiet der theoretischen Physik zu schreiben und mich auf diesem Weg mit Rat und Tat begleitet hat. Weiterhin, möchte ich mich bei der Bonn-Cologne Graduate School für ihre Unterstützung und bei Prof. Hans-Peter Nilles, der mir während dieser Zeit als Mentor zur Seit stand, bedanken. Mein Dank gilt auch allen Mitgliedern der Arbeitsgruppe, die mir durch die vielen Diskussionen und Gespräche eine angenehme Arbeitsumgebung und eine erfolgreiche Zusammenarbeit ermöglicht haben: Dr. Thomas Grimm, Tae-Won Ha, Prof. Albrecht Klemm, Denis Klevers, Marco Rauch, Dr. Piotr Sulkowski und Thomas Wotschke. Des weiteren bedanke ich mich bei Thomas Grimm, Denis Klevers, Marco Rauch und Thomas Wotschke für das Korrekturlesen. Mein besonderer Dank gilt meiner Familie, meiner Schwester Bita, und meinen Eltern Diana und Mohammad, für ihre wundervolle Unterstützung und Liebe. 4

5 Contents 1 Introduction Motivation Outline The Physics of the Topological String Compactifications to N = 2 supergravity Type IIA / IIB string theory in ten dimensions Compactifications to four dimensions BPS states Gauge theories from geometry The Seiberg-Witten model Gauge theories from local Calabi-Yau manifolds Counting the entropy of Black Holes Black Holes in four and five dimensions Microscopic interpretation of the entropy The Topological String The background geometry Calabi-Yau manifolds The Moduli Space Supersymmetric nonlinear Sigma Models N = (2, 2) nonlinear Sigma Models Linear Sigma Model view point Twisting the N = (2, 2) theories Generalities about topological field theories The A- and B-twists Physical Observables of the topological theories Metric (in)dependence and topological string theory Dependence on the parameters The tt equations The topological A-model A model without worldsheet gravity Coupling to topological gravity i

6 ii CONTENTS Target space perspective Interpretation around the Conifold singularity The topological B-model B-model without worldsheet gravity Picard-Fuchs equations Coupling to topological gravity The holomorphic anomaly equations Mirror Symmetry Implications for the Topological String Solving the holomorphic anomaly equations The holomorphic limit Direct Integration The holomorphic ambiguity Grassmannian Calabi-Yau backgrounds Calabi-Yau complete intersections in Grassmannians Topological invariants of the manifolds Plücker embedding Mirror Construction Picard-Fuchs equations for one-parameter models The Grassmannian Calabi-Yau (G(2, 5) 1, 1, 3) Picard-Fuchs differential equation and the structure of the moduli space g = 0 and g = 1 Gopakumar-Vafa invariants Higher genus free energies Other Models (G(2, 5) 1, 2, 2) (G(3, 6) 1 6 ) (G(2, 6) 1, 1, 1, 1, 2) (G(2, 7) 1 7 ) d black hole entropy Summary of the models Local Calabi-Yau backgrounds Local Mirror Symmetry The local A-model The local B-model Direct Integration in local Calabi-Yau geometries K P 1 P 1 = O( 2, 2) P1 P Review of the moduli space M Solving the topological string on local F 0 at large radius Solving the topological string on local F 0 at the conifold locus Solving the topological string on local F 0 at the orbifold point Relation to the family of elliptic curves K F1 = O( 2, 3) F Solving the topological string on local F 1 at large radius Solving the topological string on local F 1 at the conifold locus Relation to the family of elliptic curves ii

7 CONTENTS iii 6 K3 fibrations Calabi-Yau hypersurfaces in toric varieties Picard-Fuchs equations and the B-model Moduli Space of K3 Fibrations K3 Fibrations The Moduli Space of the Mirror Physical boundary conditions The Strong coupling singularity The weak coupling divisor and meromorphic modular forms The Seiberg-Witten plane The Gepner point Solution of the Models M 1 = P (1,1,2,2,2) 4 [8] Conclusions 147 A Yukawa-couplings from Picard-Fuchs operators 151 B Modular anomaly versus holomorphic anomaly 153 B.1 PSL(2, Z) modular forms C Details: Grassmannian Calabi-Yau manifolds 157 C.1 Chern classes and topological invariants C.2 Tables of Gopakumar-Vafa invariants C.3 5D black hole asymptotic D Details: Local Calabi-Yau manifolds 167 D.1 Gopakumar-Vafa invariants of local Calabi-Yau manifolds E Details: K3 Fibrations 171 E.1 Gopakumar-Vafa invariants

8 iv CONTENTS iv

9 Chapter 1 Introduction 1.1 Motivation One of the major problems of modern theoretical physics, both conceptually as well as practically, is the reconcilliation of the two pillars of 20th century physics, namely gravitation and quantum mechanics. The theory of gravitation is described in an astonishingly elegant way by Einstein s theory of general relativity and has thus far passed a few but impressive experimental tests. These tests are mainly of astrophysical and cosmological nature, the most prominent of which being the manifestation of the dynamical nature of space-time through Hubble s observation of an expanding universe. The conceptual core of general relativity in the context of the dialectic development of physical theories is the unification of Newton s theory of gravitation with special relativity, being itself a synthesis of Maxwell s electromagnetism and the concept of inertial systems. On the other hand the history of quantum mechanics has taken a different road on the landscape of physical theories, having it s origin in atomic physics as well as the physics of radiation and light. This time it was not a contradiction between physical theories which led to the birth of the new theory but rather a contradiction between classical theories compared to experimental observations about the radiation of black bodies and the spectrum of light emitted by individual atoms. Quantum mechanics found its climax in the development of quantum field theory which is capable of explaining the interactions between individual particles to an accuracy unprecedented among the predictive power of human theories. Today we are standing in front of the conceptual incompatibility between quantum field theory and general relativity. As a dynamical classical theory general relativity admits a Lagrangian formulation and its dynamical variables expanded around a classical solution can be interpreted as quantum fields giving rise to a spin two particle known as the graviton. However, it turns out that this quantum field theory is ill defined in the sense that it is not renormalizable. Here, we should point out that within the so called asymptotic safety program, a lot of efforts are being devoted to establishing the existence of an ultraviolet fixed point at which Quantum Einstein Gravity can be renormalized. Up to now the results of these efforts are still speculative. Nonrenormalizability of general relativity makes practical calculations concerning the quantum nature of space-time at 1

10 2 CHAPTER 1. INTRODUCTION the time right after the big bang and near black holes impossible from the beginning. Furthermore, it is known from the work of Hawking and Bekenstein, that a black hole is a thermodynamical object with an entropy which goes linearly with the area of its horizon. However, there is no way to look behind the horizon of a black hole, even theoretically, to find out what the microscopic states are which give rise to the macroscopic entropy and radiation. These issues can only be addressed in a theory of quantum gravity where the interactions of gravitons with the microscopic objects forming the black hole are well described. Switching the point of view to the one of contemporary particle physics we find similar difficulties in explaining the observed richness of particle spectra and interactions. One of the major problems of the Standard Model of particle physics are the large quantum corrections contributing to the square of the Higgs mass, known as the hierarchy problem. One way to solve it is to introduce a new symmetry, namely supersymmetry, which doubles the particle spectrum and thus provides a mechanism to cancel the quadratic divergencies appearing in loop corrections to the Higgs mass. There are also other problems like the explanation of the amount of dark matter in the universe as well as gauge coupling unification at the GUT scale, for which solutions can be found within the supersymmetric extension of the standard model. String theory is a modern theoretical approach which incorporates both, quantum gravity as well as supersymmetry. It regularizes field theory by introducing a new scale, known as the string scale, and it naturally incorporates gravitation. Furthermore, it is a supersymmetric theory providing the framework for constructing supersymmetric models. In string theory the one-dimensional trajectory of a particle in spacetime is replaced by a two-dimensional orbit of a string denoted by worldsheet. In mathematical terms the worldsheet is a Riemann surface, i.e. a complex manifold with complex dimension one. The situation is very similar to the quantum mechanical case where the introduction of Planck s constant is responsible for passing from classical to quantum physics. In a similar manner, in string theory one introduces a new fundamental constant α (10 32 cm) 2 being a parameter for the tension of the string. It turns out that this way gravity is regularized as it is no longer possible to probe spacetime beneath distances of order α cm. In other words there is an absolute minimum uncertainty in length. Classical field theory results and in particular general relativity arise then in the limit α 0. However, it turns out that string theory is only consistent, i.e. anomaly free, in a spacetime with ten dimensions. This makes it unavoidable to compactify the theory on a six dimensional compact space to establish contact with our observed four dimensional world. In order to preserve the amount of supersymmetry relevant for phenomenology the compactification manifold has to satisfy several conditions, namely it must be complex and Kähler and allow for one covariantly constant spinor. Although being very stringent, these conditions lead to a vast number of solutions known as Calabi-Yau manifolds. Each such space leads to a different four dimensional particle content and interactions and thus to a different physical vacuum state. It is conjectured that all these vacua are connected through the notion of extremal transitions [1]. The major challenge of string phenomenology is then to find the right vacuum describing our universe at the current state of evolution. On the other hand string theory also suffers from some drawbacks 2

11 1.1. MOTIVATION 3 from the conceptual point of view. It is known that there is not only one string theoretic construction but there are five consistent theories at once, all being known only in their perturbative regimes. But since 1995 it has become clear that all five theories are connected through a chain of dualities and to an eleven-dimensional theory called M theory which is only known in its low energy limit as eleven dimensional supergravity. This picture involves the existence of new extended objects known as Dp-branes which are p-dimensional analogs of the string but different in some important properties. The best understood duality is the so called mirror symmetry which relates type IIA string theory compactified on a Calabi-Yau manifold M to type IIB string theory on a mirror Calabi-Yau W. In the extremely simplified case where M is a circle of radius R, W would be a circle of radius α /R. The symmetry states that the two theories compactified in such a way admit the same particle spectrum and four dimensional physics. We will make extensive use of mirror symmetry in this thesis. The other symmetry we will be needing for our calculations is the so called heterotic - type II duality which is a symmetry of a completely different nature. Here the string coupling constant g s in the one theory which is a field theoretical quantity is related to the size of a sphere in the other theory which is a purely geometric quantity. To test such dualities BPS states become very important as these are the only states in the theory which prevail and are protected against corrections even in the nonperturbative regime. One of the major breakthroughs of string theory in recent years is the calculation of the Bekenstein-Hawking entropy for supersymmetric black holes in terms of a string theoretic microscopic description. Such an entropy calculation is possible for black holes which are extremal in the sense that their charge and their mass are in a fixed relation. On the string theory side this involves a counting of BPS states which are realized as D-branes wrapping certain cycles of the compactification manifold. Another important line of research in string theory is the construction of four dimensional supersymmetric vacua by choosing a certain compactification geometry. This involves calculating the four dimensional effective action together with its superpotential and prepotential. There are several tools available in string theory to address the above questions, each of which emphasizing a different viewpoint. One of these tools is the topological string. It is a simplified version of the critical string theory in which the path integral localizes on the topological subsector of the theory. Physically this is a simplified approximation. Within the sigma model representation of the critical string the path integral is an integral over all possible maps of the two dimensional worldsheet to the target space which in general is a complex three dimensional Calabi-Yau manifold tensored with Minkowski spacetime. Whereas in the topological string the space of integration is reduced to the space of the distinct classical solutions. One may ask why one should look at such a simplified version of the original theory. For this question there are two main answers. First of all in its standard formulation string theories exhibit an infinite number of fields and particles, an infinite dimensional symmetry algebra which is only very vaguely understood and where a lot of these symmetries are broken. There is a claim that the topological version is another phase of the physical theory, in which much more symmetries are preserved 3

12 4 CHAPTER 1. INTRODUCTION and unbroken and in which the spectrum is considerably simpler. Therefore one hopes that from the examination of the one theory one gets clues about some principals of the physical theory. The other important advantage of the topological string is that it is able to compute certain physical amplitudes of the real string theory. These are terms of the effective four-dimensional theory which depend holomorphically on the moduli. The most important examples of such terms are the superpotential, the prepotential and the gauge kinetic function of N = 1 and N = 2 supersymmetric field theories. Furthermore, topological string theory counts naturally certain BPS states and is therefore ideally suited for computing the microscopic entropy of extremal spinning five dimensional black holes. Within the OSV conjecture [2] there are clues that the theory is also of great relevance for the entropy of four dimensional supersymmetric black holes. There are two main approaches present for solving the topological string on Calabi- Yau backgrounds. The first approach is called the topological A model and has also great relevance for the mathematical point of view. In the A model the classical solutions around which the string path integral localizes are holomorphic maps from the Riemann surface of the world sheet to curves of the Calabi-Yau target space. These are instantons labeled by the genus g of the holomorphically embedded curve and the degrees d i which count the number of intersection with the divisors of the Calabi-Yau. The mathematical tool to calculate the number of such maps is called localization. It makes use of the fact that the ambient space, being the space in which the Calabi-Yau manifold is embedded, admits a group action (i.e. (C ) n in case of a n dimensional toric variety) in order to localize the path integral to fix points of the group action. The disadvantage of the A model is that it only provides solutions at the large volume point in moduli space. The second approach is called the topological B model and rests heavily on the use of mirror symmetry. Here one performs all calculations in the mirror Calabi-Yau knowing that there the classical solutions are just maps from the worldsheet to points of the target space which are much easier to control. Then one translates the result by mirror symmetry to the original Calabi-Yau where they can be interpreted as A model results. The calculation on the B model side makes use of the properties of the topological free energies around boundary divisors of the mirror Calabi-Yau moduli space where physical descriptions of the particle spectrum are available. Such information can consist of the number and type of particles becoming massless at the relevant divisor as well as phase transitions going through an enhanced symmetry point. In this thesis we shall follow the second approach to topological string theory, i.e. the B model, as this provides us with solutions on the whole of moduli space. 1.2 Outline This thesis is organized as follows. In chapter 2 we review the physics lying behind and the physics captured by topological string theory. That is we start with an introduction to type II supergravity where we compactify the ten dimensional bosonic actions down to four dimensions. This way it is possible to identify the four dimensional multiplets with the moduli fields of the com- 4

13 1.2. OUTLINE 5 pactification manifold. Next, we pass over to the description of gauge theories within the framework of N = 2 supersymmetry where in particular we concentrate on the Seiberg- Witten solution of the SU(2) gauge theory. Having reviewed this construction we present its embedding into string theory and comment on how the topological string captures certain nonperturbative aspects of these gauge theories. Finally we discuss four and five dimensional supersymmetric black holes, their macroscopic entropy, and their embedding into string theory together with a microscopic interpretation of the entropy. Chapter 3 is devoted to an introduction of the main ideas and calculational tools behind the topological string. This implies a presentation of complex geometry and the notion of Calabi-Yau manifolds as relevant target spaces. Here we will include a section about the moduli space of Calabi-Yau manifolds and special geometry as this is of great relevance for later discussions. Then we will pass over to present a review of supersymmetric sigma models. Here we describe N = (2, 2) world sheet supersymmetry compactifications and some details of the N = (2, 2) CFT. Furthermore, we include a section about the linear sigma model perspective which represents a unifying framework for various geometric constructions and phases in string theory. Following these discussions there will be a description of the A and B twists leading to topological theories. Finally the topological models are combined with a Calabi-Yau target space and coupled to topological gravity. Here we subdivide again between the A and the B model. Having discussed the topological A and B models we turn our attention to mirror symmetry, where we will explain the mirror map and the genus zero sector. Last but not least, we will pass over to solving the holomorphic anomaly equations. The first step is to find a recursive solution of the equations genus by genus and the second will be to explain the boundary conditions at various points in moduli space. In chapter 4, solutions of the topological string on Calabi-Yau manifolds which are complete intersections in Grassmannians are presented. These results are also published in [3]. The chapter starts with an introduction to Grassmannian varieties and the notion of Calabi-Yau complete intersections in these. Next, mirror symmetry for these spaces is reviewed. Then we pass on to present results of the solutions to the anomaly equations expanded on boundary points of the moduli space. Having derived topological amplitudes for this class of spaces up to genus 5 we turn our attention to the black hole interpretation and analyze the discrepancy between the microscopic and macroscopic entropy evaluations. The next chapter, chapter 5, analyzes the solutions to the anomaly equations on local Calabi-Yau manifolds. The results of this part were published in [4]. First we give a toric description for this class of manifolds and identify a Riemann surface as their mirror. Then the direct integration procedure for these spaces is described and the boundary information from conifold expansions is extracted. The last section clarifies the relations between topological amplitudes and in particular generators of the amplitudes with modular forms. Chapter 6 deals with the topological string on K3 fibrations. The results are published in [5]. The first section deals with the description of compact toric varieties and the construction of Calabi-Yau hypersurfaces in them. An important application of the 5

14 6 CHAPTER 1. INTRODUCTION toric description, which we shall review, is the derivation of Picard-Fuchs equations from symmetries of the ambient space. The second section is concerned with the duality of the Heterotic String with the type II string and the importance of K3 fibrations in this context. Then, we give an overview over the moduli space and identify all relevant boundary divisors. Having described the geometry of the moduli space we summarize the physical boundary conditions at the various divisors and present expansions of the free energies. The last section deals with the question of integrability on these spaces. The last chapter contains some concluding remarks and directions for the future. The appendices A, B contain supplementary material about the calculation of Yukawacouplings and definitions for SL(2, Z) modular forms. The last three appendices, namely C, D, E, contain the tables of BPS degeneracies, a plot about the macroscopic and microscopic black hole entropies for Grassmannians and a table for the topological invariants of Grassmannian Calabi-Yau three-folds. 6

15 Chapter 2 The Physics of the Topological String This chapter is meant to be an overview about the physical principles underlying the topological string and its main physical applications. To this respect we first review the compactification of type IIA/IIB string theory on Calabi-Yau manifolds. Then we pass over to the discussion of N = 2 gauge theories and how one can obtain them from the choice of the compactification geometry. Here we present the Seiberg-Witten solution and its relevance for topological string computations. Last but not least we turn to a short exposition of macroscopic and microscopic black hole physics. 2.1 Compactifications to N = 2 supergravity Type IIA / IIB string theory in ten dimensions Superstring theories are only consistent as quantum theories and anomaly free in ten spacetime dimensions. Their perturbative description is given by a supersymmetric sigma model with target space a ten dimensional manifold which usually splits up into R 1,3 M, where M is denoted by compactification space. In the limit of large volume of M and small string coupling the interacting theory captured by the sigma model reduces to supergravity, i.e. a quantum field theory. The nonperturbative sector of string theory contains solitonic states, namely the D-branes, which also admit a supergravity description in terms of p-forms. The nature of the nonperturbative description of string theory is still not clear and therefore a full description of D-branes and their bound states remains far away. However, in many phenomenological applications it is convenient to consider first the supergravity limit of string theory and include nonperturbative corrections as a second step. This said, we want to describe in the following the supergravity picture arising from string theory and its impact on four dimensional physics. We shall focus our exposition on the bosonic sector of low energy effective actions of type IIA and type IIB string theory. That is, we will be dealing with type IIA and type IIB supergravity in 10 dimensions. Both theories are N = 2 supersymmetric, the 7

16 8 CHAPTER 2. THE PHYSICS OF THE TOPOLOGICAL STRING difference being that in the one theory the gravitino multiplet has opposite chirality to the gravitino sitting in the graviton multiplet 1. Starting with the non-chiral type IIA theory, its massless spectrum comprises the metric ĝ MN, the two-from ˆB 2, the dilaton ˆφ, and a one and a three-form denoted by  1 and Ĉ3. Note that the fermionic components follow by supersymmetry. The bosonic action of this theory is given by [7] S IIA = [ 2 ˆφ e 1 2 where the fields strengths are defined as ( 1 2 ˆR 1 + 2d ˆφ d ˆφ 1 4Ĥ3 Ĥ3 ( ˆF2 ˆF 2 + ˆF 4 ˆF 4 ) + L top ], (2.1.1) ˆF 2 = dâ1, ˆF4 = dĉ3 ˆB 2 dâ1, Ĥ 3 = d ˆB 2, (2.1.2) and we will ignore the topological terms L top as they are not of particular importance for the argumentation we want to carry out. The type IIB theory is the chiral type II theory and its massless bosonic fields are given by the metric ĝ MN, the two-form ˆB 2, the dilaton ˆφ, and the zero, two and four-forms l, Ĉ 2 and Â4. Speaking in string theoretic terms, one sees that the two theories differ only in their RR sectors while their NS-NS sectors comprising the fields ĝ MN, ˆB2 and ˆφ are equal. As the RR sector contains only even forms in type IIB the action will only contain odd form field strengths. Its bosonic part is ( ) S IIB = 2 ˆφ e where the field strengths are defined as 1 2 ˆR ˆφ d ˆφ 1 Ĥ3 4Ĥ3 1 (dl dl + 2 ˆF 3 ˆF ) ˆF 5 ˆF 5 1 Ĥ3 dĉ2, (2.1.3) 2Â4 Ĥ 3 = d ˆB 2, ˆF3 = dĉ2 ld ˆB 2, ˆF5 = dâ4 1 2Ĉ2 d ˆB ˆB 2 dĉ2. (2.1.4) Compactifications to four dimensions To obtain a four dimensional physical theory the ten dimensional supergravities have to be compactified on a manifold M. One can choose the amount of preserved supersymmetry by the choice of the holonomy of the internal manifold M. The number of left supersymmetries will be equal to the number of spinors which can be chosen to be singlets under the holonomy group. A spinor is an irreducible representation of the algebra so(1, d 1) and has dimension 2 d/2 for d even and 2 (d 1)/2 for d odd. Furthermore, a spinor may be 1 The latter is known as the chiral theory and the former as the non-chiral ) 8

17 2.1. COMPACTIFICATIONS TO N = 2 SUPERGRAVITY 9 real (R), complex (C), or quaternionic (H) depending on d, see [8] for more details. The general rule is R if d = 1, 2, 3(mod 8) C if d = 0(mod 4) H if d = 5, 6, 7(mod 8). A complex representation has twice as many degrees of freedom as a real representation and a quaternionic representation has the same number of degrees of freedom as a complex representation due to constraints. To see what happens when one compactifies type II supergravity down to a lower number of dimensions replace the space R 1,d 0 1 by R 1,d 1 1 M, for M a compact space of dimension d 0 d 1. Then one has to consider how a spinor of so(1, d 0 1) decomposes under the maximal subalgebra so(1, d 1 1) so(d 0 d 1 ) so(1, d 0 1). The holonomy of M acts on this maximal subalgebra and each representation which is invariant under its action will lead a new supersymmetry in the compactified target space. For N = 2 in ten dimensions the general rule is the following Holonomy of M N in D = 4 SO(6) 8 SU(2) 4 Z 2 SU(2), SU(3) 2 Table 2.1.1: Holonomy and supersymmetry. Here N is the number of supersymmetries and D the dimension of spacetime. An example for the first type is the torus T 6, for the second case one can choose the target space to be K3 T 2, and the last case comprises any Calabi-Yau manifold. A mathematical definition of K3 surfaces and Calabi-Yau manifolds will be given in section 3.1. As our main interest lies in theories with N = 2 supersymmetry we will look at type II compactifications on Calabi-Yau manifolds in more detail. Compactification on Calabi-Yau manifolds The field content of N = 2 supersymmetry in four dimensions can be constructed from the multiplets of N = 1 supersymmetry. An N = 2 hypermultiplet is build from two chiral multiplets resulting in two complex scalars and two fermions. On the other hand a vector and a chiral superfield together give rise to an N = 2 vector multiplet, its field content being a vector, two gaugini and one complex scalar. The N = 2 graviton multiplet is the union of the N = 1 graviton and gravitino multiplets. We summarize these observations in table Let us start by compactifying type IIA on a Calabi-Yau. We want to identify the bosonic field content of the resulting theory with the bosonic fields presented in table In a reduction of a higher dimensional theory to a lower dimensional one, denoted by the term Kaluza-Klein-compactification, one expands the higher dimensional fields in terms of harmonics of the compact space in order to only keep massless modes in the 9

18 10 CHAPTER 2. THE PHYSICS OF THE TOPOLOGICAL STRING Multiplet Bosons Fermions hyper-multiplet 4 φ 2 ψ vector A µ, Φ 2 λ graviton g µν, A 0 µ 2 Ψ µ Table 2.1.2: N = 2 supermultiplets in four dimensions. The symbol φ is reserved for real scalars and Φ for complex ones. effective theory. Expanding in this spirit the ten dimensional fields Â1, ˆB2 and Ĉ3 in Calabi-Yau harmonic forms one obtains 2 Â 1 = A 0, ˆB 2 = B 2 + b i ω i, Ĉ 3 = C 3 + A i ω i + ξ A α A ξ A β A, (2.1.5) where C 3 is a three-form, B 2 a two-form, (A 0, A i ) are one-forms and b i, ξ A, ξ A are scalar fields in D = 4. Note that ω i, i = 1,, h 1,1 (M) are harmonic 2-forms and α A, β A, A = 0,, h 2,1, are harmonic three-forms of the internal manifold M. However, these are not yet all fields appearing in the four dimensional theory. There are also massless modes associated to metric deformations of the internal geometry. In the case of Calabi- Yau manifolds these are Kähler and complex structure deformations denoted by v i, i = 1,, h 1,1 and z a, a = 1,, h 2,1, as will be described in more detail in section The b i and v i combine together to form the complex fields t i = b i + iv i. These scalars together with the one-forms A i form the bosonic content of h 1,1 vector multiplets. Turning our attention to the complex structure deformations we see that the complex fields z a and the scalars ξ a, ξ a form together exactly the bosonic content of an N = 2 hypermultiplet, namely 4 scalar bosons. The remaining fields adjust themselves into the tensor and the gravitational multiplet. Next, we pass over to the compactification of type IIB theory. Again, in order to derive the massless spectrum the ten dimensional fields are expanded into Calabi-Yau harmonic forms ˆB 2 = B 2 + b i ω i, Ĉ 2 = C 2 + c i ω i, Â 4 = D i 2 ω i + V A α A U B β B + ρ i ω i, (2.1.6) where now in addition to the harmonics already present in the type IIA case, this time we also have harmonic (2, 2)-forms ω i, i = 1,, h 1,1 being dual to the ω i introduced earlier. Here one has to note that the bosonic fields D i 2 and ρ i are duals of one another and the one-form fields (V A, U A ) are related by electric-magnetic duality. Thus we only have to consider half of these fields. We choose the scalar fields ρ i, 1 = 1,, h 1,1, and the vector fields V A, A = 0,, h 2,1, to be physical. Combining these with the Calabi-Yau moduli we see that one is left with h 1,1 hyper-multiplets (ρ i, v i, b i, c i ) and h 2,1 vector multiplets (V a, z a ). 2 Such a reduction was performed the first time in [9] 10

19 2.1. COMPACTIFICATIONS TO N = 2 SUPERGRAVITY 11 Comparing with the type IIA spectrum in four dimensions we see that the number of hyper- and vector multiplets are exchanged. This observation is the supergravity origin of mirror symmetry. In the full string theory picture these two moduli spaces receive quantum corrections and thus make mirror symmetry a far more nontrivial statement. Looking at the vector moduli spaces, the type IIA side gets quantum corrected by worldsheet instantons while the type IIB vector moduli space remains uncorrected. Worldsheet instantons can be interpreted as BPS states arising from D-brane bound states which in turn have important applications to nonperturbative aspects of string theory. Indeed, in this thesis we will make extensive use of mirror symmetry to calculate the degeneracy of specific BPS states and hence we will use the next subsection to introduce them briefly to the reader BPS states BPS states are massive supersymmetric states which play an important role in the understanding of the nonperturbative nature of Superstring theory as their properties are protected against corrections even in the nonperturbative regime. Let us present their definition and their main properties in a short exposition here, for more detail we refer to the lecture notes [10]. Consider a theory with N supersymmetries where N = 2r for some r. Diagonalizing the anti-symmetric central charge matrix Z ab = Z ba of the theory into blocks of 2 2, we obtain Z = diag(ɛz 1,, ɛz r ) ɛ 12 = ɛ 21 = 1, (2.1.7) where the Zā, ā = 1,, r are called the real central charges. Next, we look at the massive representations of the theory and define creation and annihilation operators by Qāα± 1 2 (Q1ā α ± σ 0 (Q2ā α β β ) ) and their hermitian conjugates. The only nontrivial supersymmetry algebra relation left is then } {Qāα±, (Q bβ± ) = δā b δα(m β ± Zā). (2.1.8) The left hand side (2.1.8) must be positive for any unitary representation of the supersymmetry algebra. This immediately gives us the so called BPS bound for the mass of the particles in the spectrum M Zā ā = 1,, r = [N/2]. (2.1.9) For configurations with Zā = M the BPS bound is saturated and one of the supercharges Qāα+ or Qāα must vanish. As a consequence we obtain a shorter supersymmetry representation, i.e. the phenomenon of multiplet shortening occurs. If M = Zā for ā = 1,, r 0, and M > Zā for other values of ā, the corresponding supersymmetry representation has dimension 2 2N 2r 0 and is denoted by 1/2 r 0 BPS. For N = 2 supersymmetry in four dimensions we list the number of irreducible spin representation for BPS saturated multiplets in table

20 12 CHAPTER 2. THE PHYSICS OF THE TOPOLOGICAL STRING spin 1 0 1/2 1 N = 2 BPS hyper N = 2 BPS vector Table 2.1.3: Number of irreducible representations as a function of spin The names hyper and vector arise from state counting which shows that they have the same number of states as massless hyper- and massless vector multiplets. The condition M = Zā will remain valid even in the strong coupling regime and will not suffer corrections as one does not expect short multiplets to turn into the full multiplets, with many more states! 2.2 Gauge theories from geometry Having described four dimensional supergravity we turn next to supersymmetric gauge theories. We will see that in string theory there exists a mechanism to decouple gravity from these theories and thus obtain a purely gauge theoretic description. In the following we will first describe the Seiberg-Witten gauge theory as it gives rise to many interesting features, and as a second step we will explain its embedding into string theory The Seiberg-Witten model The setup Consider N = 2 supersymmetric Yang-Mills theory in four dimensions. Assume that the gauge theory is SU(2) with one vector supermultiplet A. Then the particle content of A is, according to section (2.1.2), given by an N = 1 chiral multiplet, whose components we shall denote by φ and ψ, and an N = 1 vector multiplet with components λ and one gauge field A µ. All fields come in the adjoint representation. Under the global SU(2) R symmetry the bosonic fields A µ and φ are singlets and λ, ψ form a doublet. Furthermore, there is an additional U(1) R symmetry acting on the fields φ,ψ. However, quantum mechanically this R-symmetry is broken to its Z 8 subgroup by an anomaly in the theory we are considering. In N = 1 superspace formalism, the Lagrangian is expressed as [ 1 4π Im d 4 θ F(A) A A + d 2 θ 1 2 ] 2 F(A) W A 2 α W α, (2.2.1) where A is the N = 1 chiral multiplet in the N = 2 vector multiplet A, and its scalar component we denote by a. The prepotential F gives rise to the Kähler potential ( ) F(A) K = Im A A. (2.2.2) 12

21 2.2. GAUGE THEORIES FROM GEOMETRY 13 Note that we have not included a superpotential, but the D-term gives rise to the following classical scalar potential V (φ) = 1 g 2 Tr[φ, φ ] 2. (2.2.3) So, classically the vacua of the theory are given by the configurations where φ and φ commute. In our case the gauge group is SU(2) and thus we can take φ = 1 2 aσ3, with σ 3 = diag(1, 1) and a a complex parameter. As SU(2) acts by its Weyl group on the field a, sending it to a, we see that the gauge inequivalent vacua are parametrized by the gauge invariant quantity u = 1 2 a2 = Trφ 2. For non-zero a supersymmetry remains unbroken while the gauge symmetry is broken to U(1) and the global Z 8 symmetry is broken to Z 4. The solution Seiberg and Witten [11] have presented a solution to the effective infrared limit of the theory, described by the Wilsonian action. That is, they have constructed the space of effective quantum corrected gauge inequivalent vacua and have deduced the particle spectrum from it. Their Ansatz relies on three major considerations holomorphy, global symmetries, the existence of a nonsingular weak coupling limit. As all quantities of interest can be deduced from the holomorphic prepotential F, the goal will be to compute its full quantum corrected expansion. The perturbative corrections to F were already deduced in [12]. The tree level and one loop contributions add up to F oneloop = i 1 2π A2 ln A2 Λ, (2.2.4) where Λ is the dynamically generated scale and all higher loop contributions vanish. The logarithm in the expression is responsible for the anomalous transformation behavior of the U(1) R. Further corrections arise from instantons. The new terms have to be invariant under the remaining Z 4 R-symmetry which suggests the holomorphic Ansatz F = i 1 2π A2 ln A2 Λ + ( 4k Λ F 2 k A A) 2, (2.2.5) where the k th term arises as a contribution of k instantons. Negative powers of k are absent as they would violate the existence of a nonsingular weak coupling limit. It will turn out that infinitely many of the F k are nonzero. Seiberg and Witten deduce the instanton corrected prepotential by looking at the metric on the space of vacua (the 13 k=1

22 14 CHAPTER 2. THE PHYSICS OF THE TOPOLOGICAL STRING moduli space) and computing its behavior around specific points in moduli space. The metric is given by (ds) 2 = Imτ(a)dadā, (2.2.6) where τ(a) is a holomorphic function τ = 2 F/ a 2. The first major observation is that Imτ cannot be a globally defined smooth function as it would cease to be positive definite. As such it must have singularities on the moduli space leading to monodromies around singular points. A correct description of the metric uses the variables a D = F/ a and a, in terms of which we have (ds) 2 = Imda D dā = i 2 (da Ddā dadā D ). (2.2.7) If we parameterize the moduli space by a variable u (corresponding to Trφ 2 ), the functions a and a D will transform nontrivially by going around singular points on the u-plane. Indeed one can show that these monodromy transformations form a subgroup of SL(2, Z) denoted by Γ 2. What is the origin of these monodromies and how can one compute them? The answer lies at the heart of the physics. Seiberg and Witten conjecture that the singularities come from massive particles of spin 1/2 that become massless at particular points in the moduli space. Moreover, these particles are not elementary but bound states and correspond to monopoles and dyons. These are BPS states whose mass is given by the formula M = 2 n m a D + n e a. Let us look at a monopole with mass a D which is becoming massless at the point a D (u 0 ) = 0. Then the Wilsonian effective action will incorporate the effect of integrating out such a massless particle. More precisely, using the one loop beta function one can show that the magnetic coupling is τ D i π lna D. (2.2.8) Using this result and the relation between the functions a and a D described in [11] one can show that by going in a loop around the point u 0 in the u-plane a and a D transform as a D a D (2.2.9) a a 2a D. (2.2.10) A similar behavior shows up around two other points in moduli space: the massless dyon point and the weak coupling point u =. The picture unraveled here is the one of a Riemann surface with periods a and a D! Using the symmetries of the theory and the details of the monodromy behavior just described this Riemann surface can be identified to be described by the equation y 2 = (x 1)(x + 1)(x u). (2.2.11) The singular points x = 1 and x = 1 correspond the massless monopole and the massless dyon point. This beautiful picture finally solves the initial problem as deducing the prepotential associated to the moduli space of this Riemann surface is equivalent to computing the instanton corrected holomorphic function F of the N = 2 theory. 14

23 2.2. GAUGE THEORIES FROM GEOMETRY 15 Figure 2.1: The Seiberg-Witten u-plane with a choice of base point Gauge theories from local Calabi-Yau manifolds The Seiberg-Witten SU(2) gauge theory can be embedded into string theory in the sense that it can be obtained in a certain limit from a Calabi-Yau compactification. This was analyzed in [13]. The main idea can be traced back to the observation that in type IIA compactifications over K3 fibrations (a K3 surface fibred over P 1 = S 2 ) ADE singularities of K3 lead to enhanced gauge symmetry of ADE type [14]. The reason is that 2-branes of type IIA wrapped around vanishing 2-cycles lead to precisely the missing states expected for gauge symmetry enhancement. In the case of the SU(2) theory the geometry of the Calabi-Yau will consist of a base P 1 with fibre in the singular limit being C 2 /Z 2. Blowing up C 2 /Z 2 we see that the geometry locally contains a fibration of P 1 over P 1. The W ± will correspond to 2- branes wrapped around the P 1 fibre and their mass is proportional to the area of the 2-sphere. Furthermore, 1/g 2 is proportional to the area of the base sphere, where g is coupling constant of the gauge theory. Now, the nature of the particular limit taken in [13] is sending M planck in order to decouple gravity and obtain a pure gauge theory. Geometrically this is realized by sending size of the base, denoted by t b, to infinity, i.e. t b, and the size of the P 1 fibre, denoted by t f, to zero. However, as we have exp( 1/g 2 ) = exp( t b ) ɛ 4 Λ 4 t f ɛa, (2.2.12) we have to ensure that the ratio exp( 1/g) stays finite in order to obtain the finite instanton t 4 f contributions in (2.2.5) while sending ɛ 0. Of what use is the topological string here? The genus 0 topological free energy, denoted by F 0, encodes all instanton corrections of the type IIA vector moduli space. Due to mirror symmetry it can be computed classically on the mirror manifold and then translated back to the type IIA side. As it turns out, in the local limit we are employing here the mirror manifold is characterized by a Riemann surface which turns out to be exactly the same as the Seiberg-Witten curve (2.2.11). Therefore, the genus 0 topological string free energy contains all instanton contributions of the gauge theory. This result can be used to obtain systematically nonperturbative corrections to quantum field theories. 15

24 16 CHAPTER 2. THE PHYSICS OF THE TOPOLOGICAL STRING A further generalization of the setup just presented, apart from going to higher gauge groups, is to couple the gauge theory to matter. This is done by including hypermultiplets in the adjoint or in the fundamental representation. Geometrically the inclusion of r hypermultiplets in the adjoint is engineered by fibering ADE singularities over a complex curve of genus r (see figure 2.2). Figure 2.2: Illustration of an N = 2 SU(2) gauge theory with 3 hypermultiplets in the adjoint. 2.3 Counting the entropy of Black Holes Now we shall leave the path of pure gauge theories and return to the supergravity picture. This is the correct arena for analyzing the theory of black holes. From the point of view of the topological string this means including the higher free energies F g (t i ) as these correspond to the following F-terms in the effective four dimensional N = 2 supergravity d 4 xf g (t i )R 2 +F 2g 2 +. (2.3.1) Here, R + is the self-dual part of the Riemann tensor and F + is the self-dual part of the graviphoton field strength. The couplings F g (t i ) depend on the vector moduli arising from compactification of type IIA string theory on a Calabi-Yau manifold. The general rule to compute the F g is very similar to the one used in the Seiberg-Witten solution. The terms (2.3.1) arise in the Wilsonian effective action by integrating out massive states. However, on certain points on the moduli space of the Calabi-Yau manifold some of the states integrated out become massless and lead to singularities in the effective four dimensional theory. In the case of the topological string these are BPS states corresponding to D-brane bound states. A knowledge of the expansion of the F g (t i ) around singular points in the moduli space and of the monodromy of the periods there can be restrictive enough to fix the F g completely. This has important applications for the theory of black holes to which we shall turn next. 16